Inclined Plane

From the pyramids to motorways

The inclined plane, or ramp, is one of humanity's oldest tools. When the pyramids were being built in Egypt (around 2600 BC), the huge stone blocks are thought to have been dragged up ramps. Ramps are still everywhere today: wheelchair ramps, motorway slip roads, truck loading ramps, mountain roads. Even the thread of a screw is an inclined plane wrapped around a cylinder.

The inclined plane is one of the six classical simple machines, along with the lever, the pulley, the wheel and axle, the wedge and the screw.

Splitting the weight in two

The weight (W = mg) of an object on a ramp always points straight down. But to understand how the object behaves on the ramp, we split this force into two parts relative to the ramp:

Component	Size	Direction
W∥	mg·sinθ	Down along the ramp
W⊥	mg·cosθ	Perpendicular to the surface, into the surface

Because these two components are perpendicular to each other, together they give the weight through the Pythagorean relation:

$W_∥^2 + W_⊥^2 = (mg)^2$

The weight does not change; it is only divided into two different effects relative to the ramp. In the interactive model the dashed orange arrows show these two components. As you make the ramp steeper, W∥ grows and W⊥ shrinks.

Normal force: The surface pushes back on the object. This force (N) is perpendicular to the surface, points outward, and is equal in size to W⊥: $N = mg\cos θ$ . The steeper the slope, the less the object presses into the surface.

Friction: two kinds

Static friction stops a resting object from sliding. It grows as much as needed, but can reach at most:

$f_{s,\text{max}} = \mu_s \cdot N = \mu_s \cdot mg\cos θ$

Kinetic friction slows down a sliding object. Its size is constant, and it is usually smaller than the maximum static friction:

$f_k = \mu_k \cdot N$

That is why getting a heavy piece of furniture moving is harder than keeping it moving.

When does the object slide?

A resting object starts to slide when the force pulling it down exceeds the maximum static friction:

$mg\sin θ > \mu_s\, mg\cos θ \quad\Rightarrow\quad \tan θ > \mu_s$

The mass cancels on both sides; the angle of sliding depends only on the surfaces. This is a simple way to measure the coefficient of friction: slowly make the ramp steeper, and the tangent of the angle at which the object starts to slide gives $\mu_s$ .

The acceleration of an object once it starts to slide is:

$a = g(\sin θ - \mu_k \cos θ)$

Mass does not appear in this formula either. Change the mass in the interactive model: all the forces grow, but the acceleration stays the same.

Pushing up the ramp

To start sliding a resting object up the ramp, you need to overcome both W∥ and static friction:

$F \geq mg\sin θ + \mu_s\, mg\cos θ$

Friction can change direction. With no push, friction points up the slope to stop the object sliding down. With a strong enough push, it points down the slope to stop the object sliding up. In the model you can slowly increase the pushing force and watch the friction arrow shrink and change direction.

Mechanical advantage and work

Without friction, the force needed to push the object up is only $mg\sin θ$ . Compared with lifting it straight up, the gain is:

$\text{MA} = \frac{mg}{mg\sin θ} = \frac{1}{\sin θ} = \frac{L}{h}$

For example, on a 30° slope MA = 2: you can move the object up with a force equal to half its weight, but you have to travel twice the height along the ramp. The work done ( $F \cdot L = mg \cdot h$ ) stays the same. You gain in force and lose in distance.

Galileo and the inclined plane

Galileo Galilei used the inclined plane to study falling objects. Free fall was too fast to measure with the naked eye. By rolling balls along a gently sloping groove he slowed the motion down and showed that the distance travelled is proportional to the square of the elapsed time. This was one of the first measurements of motion with constant acceleration.

In the real world

Wheelchair ramps: In Türkiye, the TS 9111 standard says the slope should generally be at most 8% (about 1/12, 4.8°), close to the 1:12 limit used in many other countries. At this slope the mechanical advantage is about 12.
Motorways: So that heavy vehicles can climb hills, slopes on main roads are usually kept below 6–7% (3.5–4°).
The screw: The thread of a screw is an inclined plane wrapped around a cylinder. With each turn the screw advances only by the thread pitch; the large ratio between the circumference and the pitch turns a small turning force into a large clamping force.
The wedge: The edge of an axe or a knife is two inclined planes joined back to back.

Interesting numbers

A ramp up Everest: If you wanted to climb to 8849 m on a ramp with a 5° slope, the ramp would be $8849 / \sin 5° \approx 101$ km long.
Baldwin Street (New Zealand): at about 35% (19°), one of the steepest residential streets in the world.
Ski slopes: The steepest sections of World Cup downhill courses reach gradients of 70–85% (35–40°).
Sliding on ice: The static friction coefficient of ice is about 0.1; objects start to slide on an icy road with a slope of only $\arctan 0.1 \approx 6°$ .

Summary

In depth